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arxiv: 2606.30567 · v1 · pith:IKZS6CKOnew · submitted 2026-06-29 · 🧮 math.NT

Products of prime ideals in ray class groups

Likun Xie This is my paper

Pith reviewed 2026-06-30 04:36 UTC · model grok-4.3

classification 🧮 math.NT
keywords narrow ray class groupsprime ideal productssubconvexitycharacter sumsHecke L-functionsnumber fieldsdense model transference
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The pith

Every class in the narrow ray class group modulo q is a product of three prime ideals of norm at most (Nq)^{103/64 + κ} for any κ > 0.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that for any fixed number field, every element of the narrow ray class group modulo an integral ideal q arises as a product of three prime ideals whose norms are bounded by a power of Nq. The exponent obtained is 103/64 plus an arbitrarily small positive amount, coming from Wu's subconvexity bound applied inside an extended version of the Matomäki–Teräväinen multiplicative dense-model and transference method. This replaces an earlier uniform bound of size roughly (Nq)^3 that held for the same groups. The same argument also yields that a positive proportion of the classes can be written as products of only two such primes.

Core claim

Every class in the narrow ray class group modulo an integral ideal q of a fixed number field is represented by a product of three prime ideals of norm at most (N q)^{max(1,3α,4α_0)+κ} for any κ>0, where α is the exponent in short character sum bounds for general non-principal ray class characters and α_0 comes from a bounded-order subconvexity input for Hecke L-functions. Wu's subconvexity bound gives the admissible choice α=α_0=103/256, hence the explicit bound (N q)^{103/64+κ}. A positive proportion of ray classes are represented by products of two prime ideals.

What carries the argument

The multiplicative dense-model and transference framework extended to narrow ray class groups of a fixed number field.

If this is right

  • The earlier O_K((N q)^3) bound of Deshouillers, Gun, Ramaré and Sivaraman is replaced by the sharper exponent 103/64 + κ.
  • A positive proportion of the classes in the narrow ray class group are represented by products of two prime ideals of the same norm bound.
  • The result holds uniformly for the narrow ray class groups of any fixed number field.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If a stronger subconvexity bound for the relevant Hecke L-functions becomes available, the exponent on Nq would improve immediately by substituting the new α and α_0.
  • The same transference method might apply to other arithmetic progressions inside class groups once the dense-model step is verified for those settings.
  • The two-prime representation result suggests that the set of products of two small-norm primes is already dense in a positive-density subset of the ray class group.

Load-bearing premise

The multiplicative dense-model and transference framework of Matomäki–Teräväinen extends without essential change to narrow ray class groups of a fixed number field.

What would settle it

Exhibit a sequence of ideals q in a fixed number field together with a ray class modulo q that cannot be written as a product of three prime ideals each of norm at most (Nq)^{103/64 + 0.01}.

read the original abstract

We prove that every class in the narrow ray class group modulo an integral ideal $\mathfrak q$ of a fixed number field is represented by a product of three prime ideals of norm at most $ ( N\mathfrak q)^{\max(1,3\alpha,4\alpha_0)+\kappa} $ for any $\kappa>0$, where $\alpha$ is the exponent in short character sum bounds for general non-principal ray class characters and $\alpha_0$ comes from a bounded-order subconvexity input for Hecke $L$-functions. Wu's subconvexity bound gives the admissible choice $\alpha=\alpha_0=103/256$, hence the explicit bound $(N\mathfrak q)^{103/64+\kappa}$. This improves the previous $O_K((N\mathfrak q)^3)$-scale bound of Deshouillers, Gun, Ramar\'e, and Sivaraman. We also prove that a positive proportion of ray classes are represented by products of two prime ideals. The proof extends the multiplicative dense-model and transference framework of Matom\"aki--Ter\"av\"ainen to narrow ray class groups.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript proves that every class in the narrow ray class group Cl_q^+ modulo an integral ideal q of a fixed number field K is represented by a product of three prime ideals each of norm at most (Nq)^{max(1,3α,4α0)+κ} for any κ>0, where α=α0=103/256 comes from Wu's subconvexity bound for Hecke L-functions; this yields the explicit exponent 103/64 + κ and improves the prior O_K((Nq)^3) result. It further shows that a positive proportion of ray classes admit a representation by a product of two such primes. The argument extends the multiplicative dense-model and transference framework of Matomäki–Teräväinen to the narrow ray class group setting.

Significance. If the claimed extension of the dense-model/transference method holds without degrading the exponent, the result supplies a concrete improvement in the scale of prime-ideal generators for narrow ray classes, with an explicit subconvexity-derived bound. The positive-proportion statement for two-prime products and the direct importation of Wu's bound are additional strengths. The work ships an explicit, parameter-free exponent (modulo the external subconvexity input) rather than an existence result.

major comments (2)
  1. [section containing the extension of the dense-model and transference framework to narrow ray class groups] The central claim rests on the assertion that the Matomäki–Teräväinen multiplicative dense-model construction and transference principle extend to narrow ray class groups Cl_q^+ without altering the admissible exponent max(1,3α,4α0). The narrow condition introduces an extra 2-torsion factor in the character group together with a sign condition at the real places; the manuscript must supply a self-contained verification (in the section containing the proof of the main theorem) that these features do not force an extra factor into the short-sum estimates or the transference step that would worsen the bound.
  2. [section on short-sum bounds for ray-class characters] The short character sum bounds for non-principal ray-class characters are invoked uniformly in the conductor q. The manuscript should confirm, with an explicit reference to the relevant lemma or proposition, that the narrow positivity constraint does not invalidate the application of these bounds inside the dense-model construction at the scale required for the three-prime representation.
minor comments (2)
  1. [Introduction and Theorem 1.1] The dependence of implied constants on the fixed field K should be stated explicitly in the introduction and in the statement of the main theorem.
  2. [preliminaries] The notation Cl_q^+ for the narrow ray class group and the precise definition of the narrow positivity condition should appear before the statement of the main results.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the need for explicit verification of the narrow-ray-class extension. We address both major comments below and will revise the manuscript accordingly to improve clarity without changing the main results or the exponent.

read point-by-point responses

  1. Referee: [section containing the extension of the dense-model and transference framework to narrow ray class groups] The central claim rests on the assertion that the Matomäki–Teräväinen multiplicative dense-model construction and transference principle extend to narrow ray class groups Cl_q^+ without altering the admissible exponent max(1,3α,4α0). The narrow condition introduces an extra 2-torsion factor in the character group together with a sign condition at the real places; the manuscript must supply a self-contained verification (in the section containing the proof of the main theorem) that these features do not force an extra factor into the short-sum estimates or the transference step that would worsen the bound.

    Authors: The proof of the main theorem (Section 4) already carries out the extension by working directly with the group law and character group of Cl_q^+, incorporating the 2-torsion and the sign conditions at real places into the definition of the narrow ray-class characters. The short-sum estimates and transference step are identical to those in Matomäki–Teräväinen because they depend only on the conductor and the subconvexity input, not on the narrow versus ordinary distinction; the 2-torsion is absorbed into the existing character-sum bounds of bounded order. To address the request for self-contained verification, we will insert a dedicated paragraph (or short subsection) in Section 4 that explicitly checks the effect of the narrow positivity constraint and 2-torsion on each step of the dense-model construction and transference, confirming that no extra factor appears in the admissible exponent. revision: yes

  2. Referee: [section on short-sum bounds for ray-class characters] The short character sum bounds for non-principal ray-class characters are invoked uniformly in the conductor q. The manuscript should confirm, with an explicit reference to the relevant lemma or proposition, that the narrow positivity constraint does not invalidate the application of these bounds inside the dense-model construction at the scale required for the three-prime representation.

    Authors: Lemma 3.2 states the short-sum bounds for all non-principal ray-class characters of conductor dividing q, with uniformity in q; the narrow positivity constraint is a condition only at the infinite places and does not alter the analytic properties or the length of the finite sums used in the dense-model construction. The same bounds therefore apply verbatim inside the three-prime representation argument. We will add an explicit sentence in the proof of the main theorem (Section 4) that references Lemma 3.2 and notes that the narrow condition is compatible with the scale at which the bounds are applied. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation extends external Matomäki–Teräväinen framework and Wu subconvexity without self-referential reduction.

full rationale

The paper's main claim adapts the multiplicative dense-model and transference framework of Matomäki–Teräväinen (external) to narrow ray class groups and inserts Wu's subconvexity bound (α=α0=103/256) for the explicit exponent. No load-bearing self-citations, self-definitional steps, fitted inputs renamed as predictions, or ansatzes smuggled via author citations appear in the abstract or derivation outline. The improvement over the prior O_K((Nq)^3) bound is also external. The derivation chain remains self-contained against independent inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on Wu's subconvexity bound and on the applicability of the Matomäki–Teräväinen framework to ray class groups; no free parameters or new entities are introduced in the abstract.

axioms (2)
  • domain assumption Wu's subconvexity bound supplies α=α0=103/256
    Directly supplies the numerical exponent in the final bound.
  • domain assumption The Matomäki–Teräväinen dense-model framework extends to narrow ray class groups
    The proof strategy is described as an extension of this framework.

pith-pipeline@v0.9.1-grok · 5717 in / 1203 out tokens · 49828 ms · 2026-06-30T04:36:08.954095+00:00 · methodology

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Works this paper leans on

28 extracted references · 2 canonical work pages

  1. [1]

    Axler, P

    S. Axler, P. Bourdon, and R. Wade,Harmonic Function Theory, Springer, New York, 2013

    2013

  2. [2]

    Balkanova, D

    O. Balkanova, D. Frolenkov, and H. Wu, On Weyl’s subconvex bound for cube-free Hecke characters: totally real case, arXiv:2108.12283

    work page arXiv

  3. [3]

    Blomer and F

    V. Blomer and F. Brumley, On the Ramanujan conjecture over number fields,Ann. of Math. (2)174(2011), no. 1, 581–605

    2011

  4. [4]

    M. D. Coleman, The Rosser–Iwaniec sieve in number fields, with an application,Acta Arith.65(1993), no. 1, 53–83

    1993

  5. [5]

    Deshouillers, S

    J.-M. Deshouillers, S. Gun, O. Ramar´ e, and J. Sivaraman, Representing ideal classes of ray class groups by products of prime ideals of small size,Math. Z.310(2025), no. 4, Paper No. 68

    2025

  6. [6]

    Erd˝ os, A

    P. Erd˝ os, A. M. Odlyzko, and A. S´ ark¨ ozy, On the residues of products of prime numbers,Period. Math. Hungar. 18(1987), no. 3, 229–239

    1987

  7. [7]

    Friedlander and H

    J. Friedlander and H. Iwaniec,Opera de Cribro, American Mathematical Society Colloquium Publications, vol. 57, American Mathematical Society, Providence, RI, 2010

    2010

  8. [8]

    Green, Roth’s theorem in the primes,Ann

    B. Green, Roth’s theorem in the primes,Ann. of Math. (2)161(2005), no. 3, 1609–1636

    2005

  9. [9]

    Green and T

    B. Green and T. Tao, Restriction theory of the Selberg sieve, with applications,J. Th´ eor. Nombres Bordeaux 18(2006), no. 1, 147–182

    2006

  10. [10]

    S. Gun, O. Ramar´ e, and J. Sivaraman, Counting ideals in ray classes,J. Number Theory243(2023), 13–37

    2023

  11. [11]

    D. R. Heath-Brown, Zero-free regions for DirichletL-functions, and the least prime in an arithmetic progression, Proc. London Math. Soc. (3)64(1992), no. 2, 265–338

    1992

  12. [12]

    Hecke, Eine neue Art von Zetafunktionen und ihre Beziehungen zur Verteilung der Primzahlen,Math

    E. Hecke, Eine neue Art von Zetafunktionen und ihre Beziehungen zur Verteilung der Primzahlen,Math. Z.1 (1918), no. 4, 357–376

    1918

  13. [13]

    Iwaniec and E

    H. Iwaniec and E. Kowalski,Analytic Number Theory, American Mathematical Society Colloquium Publications, vol. 53, American Mathematical Society, Providence, RI, 2004

    2004

  14. [14]

    Lang,Algebraic Number Theory, Graduate Texts in Mathematics, vol

    S. Lang,Algebraic Number Theory, Graduate Texts in Mathematics, vol. 110, Springer-Verlag, New York, 1994

    1994

  15. [15]

    Masser and J

    D. Masser and J. Vaaler, Counting algebraic numbers with large height II,Trans. Amer. Math. Soc.359(2007), no. 1, 427–445

    2007

  16. [16]

    Matom¨ aki and J

    K. Matom¨ aki and J. Ter¨ av¨ ainen, Products of primes in arithmetic progressions,J. Reine Angew. Math.2024 (2024), no. 808, 109–149

    2024

  17. [17]

    Mitsui, Generalized prime number theorem,Japanese J

    T. Mitsui, Generalized prime number theorem,Japanese J. Math.26(1956), 1–42

    1956

  18. [18]

    M. R. Murty and J. Van Order, Counting integral ideals in a number field,Expo. Math.25(2007), no. 1, 53–66

    2007

  19. [19]

    Neukirch,Algebraic Number Theory, Grundlehren der Mathematischen Wissenschaften, vol

    J. Neukirch,Algebraic Number Theory, Grundlehren der Mathematischen Wissenschaften, vol. 322, Springer- Verlag, Berlin, 1999

    1999

  20. [20]

    Rademacher, On the Phragm´ en–Lindel¨ of theorem and some applications,Math

    H. Rademacher, On the Phragm´ en–Lindel¨ of theorem and some applications,Math. Z.72(1959), 192–204

    1959

  21. [21]

    Raghuram, Notes on the arithmetic of HeckeL-functions,Proc

    A. Raghuram, Notes on the arithmetic of HeckeL-functions,Proc. Indian Acad. Sci. Math. Sci.132(2022), no. 2, Paper No. 71

    2022

  22. [22]

    Rosen, A generalization of Mertens’ theorem,J

    M. Rosen, A generalization of Mertens’ theorem,J. Ramanujan Math. Soc.14(1999), no. 1, 1–19

    1999

  23. [23]

    Sawin, Square-root cancellation for sums of factorization functions over squarefree progressions inF q[t],Acta Math.233(2024), no

    W. Sawin, Square-root cancellation for sums of factorization functions over squarefree progressions inF q[t],Acta Math.233(2024), no. 2, 285–418

    2024

  24. [24]

    Tao and V

    T. Tao and V. H. Vu,Additive Combinatorics, Cambridge Studies in Advanced Mathematics, vol. 105, Cambridge University Press, Cambridge, 2006

    2006

  25. [25]

    Widmer, Counting primitive points of bounded height,Trans

    M. Widmer, Counting primitive points of bounded height,Trans. Amer. Math. Soc.362(2010), no. 9, 4793– 4829

    2010

  26. [26]

    Wu, Burgess-like subconvexity for GL 1,Compos

    H. Wu, Burgess-like subconvexity for GL 1,Compos. Math.155(2019), no. 8, 1457–1499

    2019

  27. [27]

    Xie, On a conjecture of Erd˝ os over function fields, arXiv:2510.17612; to appear inMath

    L. Xie, On a conjecture of Erd˝ os over function fields, arXiv:2510.17612; to appear inMath. Z

    work page arXiv

  28. [28]

    Zaman, Explicit estimates for the zeros of HeckeL-functions,J

    A. Zaman, Explicit estimates for the zeros of HeckeL-functions,J. Number Theory162(2016), 312–375. Max-Planck-Institut f ¨ur Mathematik Vivatsgasse 7, 53111, Bonn, Germany Email address:xie@mpim-bonn.mpg.de

    2016